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Theorem cnvss 4597
Description: Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
Assertion
Ref Expression
cnvss (𝐴𝐵𝐴𝐵)

Proof of Theorem cnvss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3017 . . . 4 (𝐴𝐵 → (⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ 𝐵))
2 df-br 3838 . . . 4 (𝑦𝐴𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
3 df-br 3838 . . . 4 (𝑦𝐵𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
41, 2, 33imtr4g 203 . . 3 (𝐴𝐵 → (𝑦𝐴𝑥𝑦𝐵𝑥))
54ssopab2dv 4096 . 2 (𝐴𝐵 → {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
6 df-cnv 4436 . 2 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
7 df-cnv 4436 . 2 𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
85, 6, 73sstr4g 3065 1 (𝐴𝐵𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1438  wss 2997  cop 3444   class class class wbr 3837  {copab 3890  ccnv 4427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-in 3003  df-ss 3010  df-br 3838  df-opab 3892  df-cnv 4436
This theorem is referenced by:  cnveq  4598  rnss  4653  relcnvtr  4937  funss  5020  funcnvuni  5069  funres11  5072  funcnvres  5073  foimacnv  5255  tposss  5993
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